How could I solve this chain rule problem?To find: d d t [ f (...

Pattab

Pattab

Answered

2022-07-16

How could I solve this chain rule problem?
To find: d d t [ f ( c ( t ) ) ] | t = 0
Where c ( t ) is such that d d t c ( t ) = F ( c ( t ) ), with F = f, c ( 0 ) = ( π 2 , π 2 ) and f ( x , y ) = sin ( x + y ) x y
I think I would need to proceed using the chain rule but I am currently not sure. How would I go about solving this problem?
Thanks in advance

Answer & Explanation

iskakanjulc

iskakanjulc

Expert

2022-07-17Added 18 answers

Let g = f c, i.e. let
g ( t ) = f ( c ( t ) ) = f ( c 1 ( t ) , c 2 ( t ) ) .
Then by the chain rule
g ( t ) = f x ( c ( t ) ) c 1 ( t ) + f y ( c ( t ) ) c 2 ( t ) = f ( c ( t ) ) c ( t ) = f ( c ( t ) ) f ( c ( t ) ) ,
where the second equality follows from the assumption about c ( t )
Since
f x ( x , y ) = y [ sin ( x + y ) + x cos ( x + y ) ]
and
f y ( x , y ) = x [ sin ( x + y ) + y cos ( x + y ) ]
we have
f x ( c ( 0 ) ) = f x ( π 2 , π 2 ) = π 2 4
f y ( c ( 0 ) ) = f y ( π 2 , π 2 ) = π 2 4
or
f ( c ( 0 ) ) = ( π 2 4 , π 2 4 ) .
Thus
g ( 0 ) = f ( c ( 0 ) ) f ( c ( 0 ) ) = ( π 2 4 , π 2 4 ) ( π 2 4 , π 2 4 ) = π 4 8 .

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